Gérald Gurtner

Physics, Networks and Agent-Based Models

Complex 2d Networks

The “Union Jack” Network

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This network is composed by two types of beams (e.g. two different cross section). It is always isotropic and optimal for conductivity whatever the values of the cross-sections. It is isotropic and optimal for elasticity if and only if their cross sections are in a ratio of square(2).

The “Star” Network

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This network encompasses in fact a whole class of network, because one that change the angles between the yellow beams and the red beams, thus generating different network. All these networks have three types of beams, i.e. three different cross-sections. These cross-sections needs to be finely tuned in order to achieve optimality for elasticity. Note that unlike almost all the networks presented here, the beams are not collinear by pairs at nodes.

The “Kisrhombille” Network

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This network is composed of three types of beams and takes its name from the corresponding tiling. It is always optimal and isotropic from the conductivity point of view, one only needs to fix one ratio between the cross-sections of two types of beams to have an optimal elastic material. The extra section can be fix to whatever value. This is some of “redundant” optimal geometry.

As a bonus, this network is nice because there is an optical illusion: the nodes seems to shimmer when you move your eyes…

Next: Interlude: Random Networks — The “Delaunay” Network

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